Minimal NMR distance information for rigidity of protein graphs

Improved NMR-data-compliant protein structure modeling captures context-dependent variations and expands the scope of functional inference

Nuclear magnetic resonance (NMR) spectroscopy can reveal conformational states of a protein in physiological conditions. However, sparsely available NMR data for a protein with large degrees of freedom can introduce structural artifacts in the built models. Currently used state-of-the-art methods deriving protein structure and conformation from NMR deploy molecular dynamics (MD) coupled with simulated annealing for building models. We provide an alternate graph-based modeling approach, where we first build substructures from NMR-derived distance-geometry constraints combined in one shot to form the core structure. The remaining molecule with inadequate data is modeled using a hybrid approach respecting the observed distance-geometry constraints. One-shot structure building is rarely undertaken for large and sparse data systems, but our data-driven bottom-up approach makes this uniquely feasible by suitable partitioning of the problem. A detailed comparison of select models with state-of-art methods reveals differences in the secondary structure regions wherein the correctness of our models is confirmed by NMR data. Benchmarking of 106 protein-folds covering 38-282 length structures shows minimal experimental-constraint violations while conforming to other structure quality parameters such as the proper folding, steric clash, and torsion angle violation based on Ramachandran plot criteria. Comparative MD studies using select protein models from a state-of-art method and ours under identical experimental parameters reveal distinct conformational dynamics that could be attributed to protein structure-function. Our work is thus useful in building enhanced NMR-evidence-based models that encapsulate the contextual secondary and tertiary structure variations present during the experimentation and expand the scope of functional inference.

Distance geometry and protein loop modeling

Due to the role of loops in protein function, loop modeling is an important problem in computational biology. We present a new approach to loop modeling based on a combinatorial version of distance geometry, where the search space of the associated problem is represented by a binary tree and a branch-and-prune method is defined to explore it, following an atomic ordering previously given. This ordering is used to calculate the coordinates of atoms from the positions of its predecessors. In addition to the theoretical development, computational results are presented to illustrate the advantage of the proposed method, compared with another approach of the literature. Our algorithm is freely available at https://github.com/michaelsouza/bpl.

NMR Protein Structure Calculation and Sphere Intersections

Nuclear Magnetic Resonance (NMR) experiments can be used to calculate 3D protein structures and geometric properties of protein molecules allow us to solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i − 3, i − 2, i − 1, with radii given by the atomic distances di −3, i, di −2, i, di −1, i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di −3, i should be represented as interval distances [ d _ i - 3 , i , d ¯ i - 3 , i {\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}} ], where d _ i - 3 , i ≤ d i - 3 , i ≤ d ¯ i - 3 , i {\underline{d}_{i - 3,i}} \le {d_{i - 3,i}} \le {\bar d_{i - 3,i}} . In the literature, an extension of the BP algorithm was proposed to deal with interval distances, where the idea is to sample values from [ d _ i - 3 , i , d ¯ i - 3 , i {\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}} ]. We present a new method, based on conformal geometric algebra, to reduce the size of [ d _ i - 3 , i , d ¯ i - 3 , i {\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}} ], before the sampling process. We also compare it with another approach proposed in the literature.

Systematic Exploration of Protein Conformational Space Using a Distance Geometry Approach

The optimization approaches classically used during the determination of protein structure encounter various difficulties, especially when the size of the conformational space is large. Indeed, in such a case, algorithmic convergence criteria are more difficult to set up. Moreover, the size of the search space makes it difficult to achieve a complete exploration. The interval branch-and-prune (iBP) approach, based on the reformulation of the distance geometry problem (DGP) provides a theoretical frame for the generation of protein conformations, by systematically sampling the conformational space. When an appropriate subset of interatomic distances is known exactly, this worst-case exponential-time algorithm is provably complete and fixed-parameter tractable. These guarantees, however, immediately disappear as distance measurement errors are introduced. Here we propose an improvement of this approach: threading-augmented interval branch-and-prune (TAiBP), where the combinatorial explosion of the original iBP approach arising from its exponential complexity is alleviated by partitioning the input instances into consecutive peptide fragments and by using self-organizing maps (SOMs) to obtain clusters of similar solutions. A validation of the TAiBP approach is presented here on a set of proteins of various sizes and structures. The calculation inputs are a uniform covalent geometry extracted from force field covalent terms, the backbone dihedral angles with error intervals, and a few long-range distances. For most of the proteins smaller than 50 residues and interval widths of 20°, the TAiBP approach yielded solutions with RMSD values smaller than 3 Å with respect to the initial protein conformation. The efficiency of the TAiBP approach for proteins larger than 50 residues will require the use of nonuniform covalent geometry and may have benefits from the recent development of residue-specific force-fields.